A Note on $L^{2}$ Harmonic Forms on a Complete ... - Project Euclid

TOKYO J. MATH.

VOL. 17, No. 2, 1994

A Note on

$L^{2}$

Harmonic Forms on a Complete Manifold Atsushi KASUE* Osaka City University (Communicated by Y. Maeda)

Introduction. In this note, we shall show a nonexistence result for harmonic forms with values in a vector bundle equipped with a Riemannian structure over a complete manifold. Moreover in relation with the result, we shall construct examples of harmonic mappings offinite total energy. This note is motivated by a recent paper of Elworthy and Rosenberg [5].

Vanishing theorems and growth properties for such forms (including harmonic mappings for example) have been investigated extensively by many authors from various points of views. Donnelly and Xavier [4] studied, for instance, the spectrum of the Laplacian acting on the square integrable forms on a negatively curved manifold and showed a sharp lower bound for the spectrum under a certain pinching condition on harmonic forms. An integral curvature. Their result gives in particular vanishing of identity on differential forms, (1.1) in Section 1, plays a crucial role in their paper. We $L^{2}$

remark that this formula was also obtained by Karcher and Wood [8] to study the growth properties for harmonic forms. In this note, we shall derive a consequence of the formula, which is stated in the following: THEOREM 1. Let $M$ be a complete Riemannian manifold of dimension $m$ , and let $E$ be a real vector bundle endowed with a Riemannian structure. Suppose $M$ possesses a pole (a point at which the exponential mapping induces a diffeomorphism). Then there are no nontrivial square integrable, E-valued harmonic q-forms ( $q=p$ or $m-p$) for a positive integer less than $m/2$ if the radial curvature of $M$ satisfies either of the following conditions: $o$

$p$

(1)

$K_{r}$

$-(\frac{m-p-1}{p})^{2}\leq K_{r}\leq-1$

Received May 17, 1993 Revised August 26, 1993

,

Partly supported by Grant-in-Aid for Scientific Research, The Ministry of Education, Scienoe and Culture, Japan. $\tau$

456

ATSUSHI KASUE

(2)

where

$-\frac{a}{1+r^{2}}\leq K_{r}\leq\frac{a^{\prime}}{1+r^{2}}$

$a\geq 0$

and

$a^{\prime}\in[0,1/4]$

.

are constants chosen in such a way that

$2+(m-p-1)\{1+(1-4a^{\prime})^{1/2}\}-p\{1+(1+4a)^{1/2}\}\geq 0$

;

$(m-p)\{1+(1-4a^{\prime})^{1/2}\}-(p-1)\{1+(1+4a)^{1/2}\}-2\geq 0$

.

Using Witten’s deformation of Laplacian on forms, Elworthy and Rosenberg showed, among other things, vanishing of harmonic p-forms under the assumptior that $[5_{-}^{-}$

$L^{2}$

$-(\frac{m-p-1}{p})^{2}+\epsilon\leq K_{r}\leq-1$

for some positive . In fact, it follows from this pinching condition that the lower bound of the spectrum of the Laplacian acting on square integrable p-forms is positive. To be precise, we shall prove the following: $\epsilon$

THEOREM 2. Let $M$ and $E$ be as in Theorem 1. Let be the lower bound of th‘ spectrum of the Laplacian acting on square integrable, E-valued p-forms of M. Then $\sigma_{p}$

$\sigma_{q}\geq\frac{1}{4}(m-1-p-ap)^{2}>0$

if $p